Mutually exclusive and exhaustive events

Two important structural properties of events:

• Mutually exclusive: no overlap. Cannot both happen.
• Exhaustive: cover the whole sample space. At least one must happen.

When events are both mutually exclusive and exhaustive, they form a partition of $S$.

Definitions

Mutually exclusive. Events $A, B$ are mutually exclusive iff $A \cap B = \varnothing$.

For more than two events: $A_1, A_2, \dots, A_n$ are pairwise mutually exclusive iff $A_i \cap A_j = \varnothing$ for every $i \ne j$.

Exhaustive. Events $A_1, \dots, A_n$ are exhaustive iff $A_1 \cup A_2 \cup \dots \cup A_n = S$.

Partition. Events form a partition of $S$ if they are pairwise mutually exclusive and exhaustive.

Examples

On a die:

• $\{1\}, \{2\}, \{3\}, \{4\}, \{5\}, \{6\}$ , partition of $S$ (the finest one).
• $\{1, 2\}, \{3, 4\}, \{5, 6\}$ , partition.
• $\{1, 2, 3\}, \{4, 5\}$ , mutually exclusive but not exhaustive (missing $6$).
• $\{1, 2, 3\}, \{3, 4, 5, 6\}$ , exhaustive but not mutually exclusive ($3$ is in both).

Why these matter

Probability of a partition. If $A_1, \dots, A_n$ partition $S$, then $P(A_1) + P(A_2) + \dots + P(A_n) = P(S) = 1.$

Complement. $A$ and $A'$ always partition $S$ (assuming $A \subsetneq S$). So $P(A) + P(A') = 1$.

These observations are foundational.

Worked examples

Example 1. Roll a die. Let $A = \{1, 2\}, B = \{3, 4\}, C = \{5, 6\}$. Show they form a partition.

Pairwise intersections empty. Union is $\{1, 2, 3, 4, 5, 6\} = S$. Yes, partition.

Example 2. Two dice rolled. Let $A_i$ = "sum is $i$" for $i = 2, 3, \dots, 12$. Are $A_2, \dots, A_{12}$ mutually exclusive? Exhaustive?

Each outcome has a unique sum, so the $A_i$ are mutually exclusive. Every outcome's sum is between $2$ and $12$ inclusive, so they are exhaustive. They form a partition of $S$ (of size $36$).

Example 3. Three coins tossed. Are "at least one head" and "at least one tail" mutually exclusive?

"At least one head" includes $\{HHH, HHT, HTH, HTT, THH, THT, TTH\}$ ($7$ outcomes). "At least one tail" includes $\{HHT, HTH, HTT, THH, THT, TTH, TTT\}$.

Intersection: $\{HHT, HTH, HTT, THH, THT, TTH\}$. Not empty. So they are not mutually exclusive.

(Are they exhaustive? Union $= S$. Yes , together they cover everything.)

Example 4. A bag has red, blue, green, yellow balls. Define $A_1 =$ "red", $A_2 =$ "blue", $A_3 =$ "green or yellow". Partition?

Each ball is exactly one of red, blue, green, yellow. $A_1, A_2$ disjoint; $A_3$ disjoint from both. $A_1 \cup A_2 \cup A_3 =$ all balls $= S$. Yes, partition.

Example 5. Two events $A, B$ are such that $P(A) = 0.5$, $P(B) = 0.4$, $P(A \cup B) = 0.7$. Are they mutually exclusive?

Mutually exclusive iff $P(A \cup B) = P(A) + P(B) = 0.9$. But $P(A \cup B) = 0.7 \ne 0.9$. So they are not mutually exclusive.

(Their intersection has probability $P(A) + P(B) - P(A \cup B) = 0.2$.)

Try it yourself

1. Roll a die. Are $\{1, 3, 5\}$ and $\{2, 4, 6\}$ mutually exclusive? Exhaustive?
2. Toss two coins. Are "HH" and "at least one tail" mutually exclusive?
3. Two dice. Are "doubles" and "sum even" mutually exclusive?
4. List a partition of the sample space when a die is rolled.
5. From $\{1, 2, \dots, 12\}$ are "multiple of $3$" and "multiple of $4$" mutually exclusive?
6. Two events $A, B$ have $P(A) = 1/3, P(B) = 1/4$. Find the minimum and maximum of $P(A \cap B)$.
7. Two events $A, B$ are exhaustive. Show $P(A) + P(B) \ge 1$.
8. Two events $A, B$ are such that $A \subset B$. Are they mutually exclusive?
9. Roll a die. Define events to form a partition of $S$ with three pieces.
10. From $\{H, T\}^3$ define a partition.
11. Three events have $P(A) + P(B) + P(C) = 1$. Must they form a partition? (Hint: no , they might not be disjoint.)
12. Are "even" and "prime" on a die mutually exclusive?

Pitfalls / Tricks

• "Mutually exclusive" and "exhaustive" are independent properties , a set of events can have either, both, or neither.
• A partition is a set of events that are both mutually exclusive and exhaustive.
• For a partition, the probabilities sum to $1$.
• Insight. Partitions are the natural way to decompose a sample space , they let you compute probability by adding contributions from disjoint cases.